Theoretical analysis demonstrates limit cycle uniqueness in special Liénard systems across ecological and economic models, suggesting broad applicability for predicting cyclical stability.
This paper investigates several autonomous planar systems of differential equations that can be transformed to a special class of Liénard system. In these systems, the ratio of the vector field’s divergence to the function locating the singularities is a rational function, with both the numerator and denominator being polynomials of degree at most two. It is well established that for Liénard systems of this type, no more than one limit cycle exists. The primary objective of this study is to demonstrate that this mathematical structure naturally emerges in numerous applications, particularly within ecological and economic dynamics. The cases discussed herein include models from recently the published papers, where the limit cycle problem remained unaddressed, alongside several novel models.
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Nong et al. (2026) studied this question.
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